General spin models from noncollinear spin density functional theory and spin-cluster expansion

TL;DR

This work introduces a torque-fitted spin-cluster expansion (SCE) framework that enables nonperturbative, self-consistent first-principles parameterization of general spin Hamiltonians including SOC effects. By fitting to site-resolved magnetic torques rather than energies, the method yields a data-efficient regression that recovers the full two-body exchange tensor (isotropic, anisotropic symmetric, and DMI) and facilitates micromagnetic mapping to extract helix periods. Real spherical harmonics and symmetry-adapted bases provide a stable, symmetry-consistent representation, while mean-field-based sampling enables controlled exploration of thermally disordered configurations. The approach is demonstrated on B20-type Mn1-x FexGe and Fe1-y Co yGe, reproducing composition-dependent trends and the divergence of the helix period near DMI sign changes, and revealing the importance of higher-order interactions at elevated temperatures. An open-source Julia package, Magesty.jl, implements the framework for data-efficient, first-principles spin-model parameterization and predictive materials design.

Abstract

We present a data-efficient framework for constructing general classical spin Hamiltonians from the spin-cluster expansion (SCE) combined with fully self-consistent noncollinear spin density functional theory (DFT). The key idea is to fit an SCE model to magnetic torques rather than to total energies. Because torques are site-resolved vectors, each configuration supplies many independent constraints, which makes the regression well conditioned and sharply reduces the number of DFT calculations needed, especially in large supercells. Applied to the B20-type chiral magnets ${\rm Mn}_{1-x}{\rm Fe}_{x}{\rm Ge}$ and ${\rm Fe}_{1-y}{\rm Co}_{y}{\rm Ge}$, the resulting models nonperturbatively extract the full pairwise exchange tensor (isotropic exchange, anisotropic symmetric exchange, and the Dzyaloshinskii--Moriya interaction) and predict helical spin period via a micromagnetic mapping. The composition trends and the divergence of the period near the chirality sign change are reproduced in line with experiments. Because the SCE framework is systematic, it also enables systematic analysis of interaction order; training on increasingly disordered spin configurations shows that the lowest-order model loses torque accuracy, whereas including higher-order interactions restores predictive power. These advances enable near-DFT-accurate spin models for finite-temperature magnetism and complex textures at modest data cost, while providing a systematic, extensible, and nonperturbative route to quantitative first-principles parameterization and predictive materials design. An open-source implementation is available as the Julia package, \textit{Magesty.jl}.

Figures (4)

• Figure 1: Example of a symmetry operation $\hat{g}$ acting on a basis function, where $\hat{g}$ is a counterclockwise rotation of $90^\circ$. The circles and accompanying arrows represent atoms and spin directions, respectively. (a) Original function $\Psi_{\mathcal{C}{\bm{l}}{\bm{m}}} = 4\pi Z_{1\bar{1}}(\hat{\bm{e}}_1)Z_{11}(\hat{\bm{e}}_2)$. (b) Transformed function $\hat{g} \Psi_{\mathcal{C}{\bm{l}}{\bm{m}}} = -4\pi Z_{11}(\hat{\bm{e}}_1)Z_{1\bar{1}}(\hat{\bm{e}}_3)$. For clarity, color images of $p_x$ (corresponding to $Z_{11}$) and $p_y$ (corresponding to $Z_{1\bar{1}}$) orbitals are also shown.
• Figure 2: Comparison of the convergence with respect to the number of DFT data points $N_{\mathrm{data}}$ for energy-fitted (black) and torque-fitted (red) regression. Panels (a) and (b) show the first-nearest-neighbor Fe--Fe exchange coupling $J_{01}$ and the magnitude of the DMI vector $|{\bm{D}}_{01}|$, respectively. The insets enlarge the torque-fitted regression results for $N_{\mathrm{data}} \le 100$.
• Figure 3: (a) Spin stiffness constant $A$, (b) spiralization constant $D$, and (c) helical spin period $\lambda$ in ${\rm Mn}_{1-x}{\rm Fe}_{x}{\rm Ge}$ and ${\rm Fe}_{1-y}{\rm Co}_{y}{\rm Ge}$. Results of previous theoretical works Kikuchi2016-iuGrytsiuk2019-paGayles2015-owKoretsune2018-jh and experimental works Grigoriev2013-vbTurgut2018-epSpencer2018-mhGuang2024-nc are also presented for reference.
• Figure 4: (a) Parity plot of site-resolved torque components. $x = T_{i\alpha}^{\rm DFT}$, $y = T_{i\alpha}^{\rm SCE}$, $\alpha \in \{x, y, z\}$. (b) Histogram of site-resolved magnetic-moment magnitudes of Fe $|{\bm M}_{\rm Fe}|$ at $\tau = 0.1$ (blue) and $\tau = 0.3$ (orange); bin width $0.01\ \mu_{\rm B}$. (c) $\langle \theta_i \rangle$ defined in Eq. (\ref{['eq:relangle']}) plotted against Fe moment magnitude $|\bm M_{\rm Fe}|$. (d)(e) Parity plots of Fe torques at $\tau = 0.3$ for $l_{\rm max} = 1$ and $l_{\rm max} = 2$, respectively. Points are color-coded by the deviation from the modal Fe moment magnitude, $|\bm M_{\rm Fe}| - |\bm M_{\rm Fe}|_{\rm mode}$. The modal value is $|\bm M_{\rm Fe}|_{\rm mode} = 1.265\ \mu_{\rm B}$.